# Homework Assignment #5

#### Due on paper in class, Friday day 17 April 2015

There are fourteen questions for 100 total points plus 10 extra credit. All are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second/third edition). though some are adapted as indicated. The number in parentheses following each problem is its individual point value.

• Problem 6.27 (3rd) (not in 2nd) (10): Let S = {(M): M is a TM and L(M) = {(M)} (that is, S is the set of TM's that accept their own description and nothing else). Show that neither S nor its complement is Turing recognizable.

• Exercise 7.4 (5):

• Exercise 7.6 (5):

• Exercise 7.10 (5):

• Exercise 7.12 (3rd) (not in 2nd) (5): Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = {(G, H): G and H are isomorphic graphs}. Show that ISO ∈ NP.

• Problem 7.13 (2nd)/7.14 (3rd) (10):

• Problem 7.14 (2nd)/7.15 (3rd) (5):

• Problem 7.17 (2nd)/7.18 (3rd) (5):

• Problem 7.24 (2nd)/7.26 (3rd) (10):

• Problem 7.25 (2nd)/7.27 (3rd) (10):

• Problem 7.35 (2nd)/7.36 (3rd) (10XC):

• Problem 7.44 (2nd)/7.46 (3rd) (10):

• Problem 7.54 (3rd) (not in 2nd) (10): In a directed graph, the indegree of a node is the number of incoming edges and the outdegree is the number of outgoing edges. Show that the following problem is NP-complete. Given an undirected graph G and a designated subset C of G's nodes, is it possible to convert G to a directed graph by assigning directions to each of its edges so that every node in C has indegree 0 or outdegree 0, and every other node in G has indegree at least 1?

• Problem 8.9 (10):